Consider a real-valued stochastic process such that a realisation of such process is a function where is some compact set of non-zero Lebesgue volume, like a hypercube, or all of . We call the index.
Suppose the process is described by a probability measure such that for , the process has expectation function and covariance We are concerned with ways to represent this covariance function .
I do not want to arse about with this mean function overmuch since it only clutters things up, so hereafter we will assume unless stated otherwise.
Wiener theorem: Deterministic case
This is also interesting and I wrote it up for a different project: See Wiener theorem.
Wiener-Khinchine theorem: Spectral density of covariance kernels
I found the Wikipedia introduction usually confusing. I recommend a well-written article, e.g. Abrahamsen (1997) or Robert J. Adler, Taylor, and Worsley (2016). Anyway, this theorem governs wide-sense-stationary random processes. Here wide-sense-stationary, a.k.a. weakly stationary or sometimes homogeneous, requires that
- the process mean function is constant, w.l.o.g. and
- correlation depends only on , i.e.
That is, the first two moments of the process are stationary, but other moments might do something weird. For the wildly popular case of Gaussian processes, since the first two moments uniquely determine the process, these end up being the same. In this context, the Wiener-Khintchine theorem tells us that there exists a finite positive measure on the Borel subsets of such that the covariance kernel is given
If has a density with respect to the dominating Lebesgue measure, then That is, the power spectral density and the covariance kernel are Fourier dual. Nifty.
What does this mean? Why do I care? Turns out this is useful for many reasons. It relates the power spectral density to the correlation function, and also to continuity/differentiability.
Bochner’s Theorem: stationary spectral kernels
Everyone seems to like the exposition in Yaglom (1987b), which I brusquely summarize here. Bochner’s theorem tells us that is the covariance function of a weakly stationary, mean-square-continuous, complex-valued random process on if and only if it can be represented as where is a positive and finite measure on (the Borel subsets of) If has a density with respect to the dominating Lebesgue measure, then is called the spectral density of and and are Fourier duals. This is what Robert J. Adler, Taylor, and Worsley (2016) calls the spectral distribution theorem.
This looks similar to the Wiener-Khintchine theorem, no? This one is telling us that the power spectrum represents all possible stationary kernels, i.e. we are not missing out on any by using a spectral representation. Note also that we needed to generalize this to complex-valued fields, and consider integrals over complex indices for it to make sense; the real fields arise as a special case.
Yaglom’s theorem
Some of the kernel design literature (Sun et al. 2018; Remes, Heinonen, and Kaski 2017; Kom Samo and Roberts 2015) cites a generalised Bochner-type Theorem (Yaglom 1987b), Yaglom’s Theorem, which does not presume stationarity:
A complex-valued, bounded, continuous function on is the covariance function of a mean-square-continuous, complex-valued, random process on if and only if it can be represented as
This is reassuring, but does not constrain kernel designs in an obviously useful way to my tiny monkey brain.
References
Adler, Robert J. 2010. The Geometry of Random Fields.
Adler, Robert J., and Taylor. 2007.
Random Fields and Geometry. Springer Monographs in Mathematics 115.
Bochner. 1959. Lectures on Fourier Integrals.
Hartikainen, and Särkkä. 2010.
“Kalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models.” In
2010 IEEE International Workshop on Machine Learning for Signal Processing.
Higdon. 2002.
“Space and Space-Time Modeling Using Process Convolutions.” In
Quantitative Methods for Current Environmental Issues.
Kom Samo, and Roberts. 2015.
“Generalized Spectral Kernels.” arXiv:1506.02236 [Stat].
Loynes. 1968.
“On the Concept of the Spectrum for Non-Stationary Processes.” Journal of the Royal Statistical Society. Series B (Methodological).
Priestley. 2004. Spectral analysis and time series. Probability and mathematical statistics.
Remes, Heinonen, and Kaski. 2017.
“Non-Stationary Spectral Kernels.” In
Advances in Neural Information Processing Systems 30.
Rust. 2007.
“Spectral Analysis of Stochastic Processes.” Lecture Notes for the E2C2/CIACS Summer School, Comorova, Romania, University of Potsdam.
Särkkä, S., and Hartikainen. 2013.
“Non-Linear Noise Adaptive Kalman Filtering via Variational Bayes.” In
2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP).
Sun, Zhang, Wang, et al. 2018. “Differentiable Compositional Kernel Learning for Gaussian Processes.” arXiv Preprint arXiv:1806.04326.
Wiener. 1930.
“Generalized Harmonic Analysis.” Acta Mathematica.
Yaglom. 1987a. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics.
———. 1987b. Correlation Theory of Stationary and Related Random Functions Volume I.